On end rotation for open rods undergoing large deformations

Heijden, van der G.H.M.; Peletier, MA Mark; Planquè, Robert

doi:10.1090/s0033-569x-07-01049-x

Cited by 17 publications

(21 citation statements)

References 17 publications

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“…This fact-related to the famous "belt trick" of Diracimplies a lower bound on elastic energy for rods subject to boundary conditions in which the end tangents are parallel [1]. It would be of great interest to expand these results to arbitrary boundary conditions, which would require some consideration of how to define an appropriate linking number for open rods [20,[58][59][60]].…”

Section: Further Discussionmentioning

confidence: 99%

Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

Hanna

2019

Journal of the Mechanics and Physics of Solids

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Motivated by observations of snap-through phenomena in buckled elastic strips subject to clamping and lateral end translations, we experimentally explore the multi-stability and bifurcations of thin bands of various widths and compare these results with numerical continuation of a perfectly anisotropic Kirchhoff rod. Our choice of boundary conditions is not easily satisfied by the anisotropic structures, forcing a cooperation between bending and twisting deformations. We find that, despite clear physical differences between rods and strips, a naive Kirchhoff model works surprisingly well as an organizing framework for the experimental observations. In the context of this model, we observe that anisotropy creates new states and alters the connectivity between existing states. Our results are a preliminary look at relatively unstudied boundary conditions for rods and strips that may arise in a variety of engineering applications, and may guide the avoidance of jump phenomena in such settings. We also briefly comment on the limitations of current strip models. * jhyutian@vt.edu † hannaj@vt.edu arXiv:1708.05968v2 [cond-mat.soft] 31 Jan 2018 the backbone of the behavior of wider bands. We reveal connections between various states, including higher-order unstable elastica modes and stable twisted states created by the rod's anisotropy.While the Kirchhoff equations show themselves to be a surprisingly useful tool in the analysis of strip behavior, we wish to emphasize that there is no reason to assume that such a model, which assumes that cross sections remain perpendicular to the centerline, would be appropriate for strips. On the other hand, the common assumption that transverse bending of strips is governed by the constraint of developability can lead to difficulties of its own, particularly for narrow strips, issues that we will briefly touch upon in an appendix. Until such issues are resolved, it is advantageous to employ an easily implemented rod model from which the strips inherit most, or even all, of their bifurcations. However, use of such a model should not be taken to imply that a narrow strip is equivalent to a rod.Boundary conditions like those we explore here are potentially of interest in helping to avoid violent snap-throughs of connectors, hinges, and umbilicals in flexible and deployable systems. Geometries similar to ours appear as slipping folds [3] in deployable space membranes, buckled elements in flexible electronics and robotics, and decorative streamers in childrens' toys [4]. Multi-stable structures find use in compliant mechanisms [5] at all length scales. The behavior of strips under our loading conditions is likely related to the phenomenon of lateral-torsional buckling, known to structural engineers [6].There is much prior work on the configurations of naturally straight rods. Work on the general behavior and classification of solutions includes that of Antman [7,8], Maddocks [9], Nizette and Goriely [10], and Cognet and co-workers [11]. Neukirch and Henderson made a detailed investigation...

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Section: Further Discussionmentioning

confidence: 99%

Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

Hanna

2019

Journal of the Mechanics and Physics of Solids

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“…(For the present case of clamped-clamped configurations this end-rotation R is closely related to the topological link Lk, see e.g. [16].) As the tension t is decreased under a threshold value t 3.7, the rod buckles and the post-buckling regime first involves 2D configurations, this is the path Ax introduced in [13].…”

Section: Bifurcation Diagrammentioning

confidence: 91%

Elastocapillary Coiling of an Elastic Rod Inside a Drop

Elettro

Grandgeorge

Neukirch

2016

J Elast

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Capillary forces acting at the surface of a liquid drop can be strong enough to deform small objects and recent studies have provided several examples of elastic instabilities induced by surface tension. We present such an example where a liquid drop sits on a straight fiber, and we show that the liquid attracts the fiber which thereby coils inside the drop. We derive the equilibrium equations for the system, compute bifurcation curves, and show the packed fiber may adopt several possible configurations inside the drop. We use the energy of the system to discriminate between the different configurations and find a intermittent regime between two-dimensional and three-dimensional solutions as more and more fiber is driven inside the drop.

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“…For the proper definition of the writhe of an open curve (where at both ends of the curve the tangents are parallel to each other), one has to complete the open curve by a closure at infinity which makes the combined curve a closed curve (see, e.g., Figure 1). We have to distinguish two classes of homotopically equivalent curves: one where two ends of the curve are rotated to each other by 2 and the other where the two ends are rotated by 4 [6,7]. This last class includes the straight line (i.e., the ground state for the spin configuration).…”

Section: Methodsmentioning

confidence: 99%

“…The first class of curves cannot be transformed continually into the ground state without crossing the curve itself. For this reason, we have to keep a memory of the triad defined on the ribbon [6]. These two classes are due to the fact that group…”

Section: Methodsmentioning

confidence: 99%

New Topological Configurations in the Continuous Heisenberg Spin Chain: Lower Bound for the Energy

Dandoloff

2015

Advances in Condensed Matter Physics

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In order to study the spin configurations of the classical one-dimensional Heisenberg model, we map the normalized unit vector, representing the spin, on a space curve. We show that the total chirality of the configuration is a conserved quantity. If, for example, one end of the space curve is rotated by an angle of 2πrelative to the other, the Frenet frame traces out a noncontractible loop inSO(3)and this defines a new class of topological spin configurations for the Heisenberg model.

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On end rotation for open rods undergoing large deformations

Cited by 17 publications

References 17 publications

Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

Elastocapillary Coiling of an Elastic Rod Inside a Drop

New Topological Configurations in the Continuous Heisenberg Spin Chain: Lower Bound for the Energy

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